Question 1

Evaluate by interpreting it in terms of areas. $\int_{-4}^{4} \sqrt{16-x^{2}} ~~d x$

Accepted Answer

The answer of Evaluate by interpreting it in terms of...

Question 2

Estimate the area from 0 to 5 under the graph of $f(x)=25-x^{2}$ using five approximating rectangles and right endpoints.

Accepted Answer

A)  $65$ 
B)  $60$ 
C)  $170$ 
D)  $270$ 
E)  $70$ 
A)  $65$ 
B)  $60$ 
C)  $170$ 
D)  $270$ 
E)  $70$

Question 3

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h(x)=\int_{1}^{e^{x}} 2 \ln t d t$

Accepted Answer

The answer of Use Part 1 of the Fundamental Theorem...

Question 4

If $f(x)=\sqrt{x}-3,1 \leq x \leq 6$ , find the Riemann sum with n = 5 correct to 3 decimal places, taking the sample points to be midpoints.

Accepted Answer

The answer of If $f(x)=\sqrt{x}-3,1 \leq x \leq 6$ ,...

Question 5

Evaluate the integral. $\int_{0}^{1} \frac{e^{2 x}}{1+e^{4 x}} d x$

Accepted Answer

The answer of Evaluate the integral. \(\int_{0}^{1} \frac{e^{2 x}}{1+e^{4 x}}...

Question 6

The area of the region that lies to the right of the y-axis and to the left of the parabola $x=6 y-y^{2}$ (the shaded region in the figure) is given by the integral $\int_{0}^{5}\left(6 y-y^{2}\right) d y$ . Find the area..

Accepted Answer

A)  $17.5$ 
B)  $12.5$ 
C)  $10.25$ 
D)  $\frac{275}{6}$ 
E)  $\frac{25}{6}$ 
A)  $17.5$ 
B)  $12.5$ 
C)  $10.25$ 
D)  $\frac{275}{6}$ 
E)  $\frac{25}{6}$

Question 7

Evaluate the definite integral. $\int_{0}^{\pi / 8} \sin~ 5 t ~d t$

Accepted Answer

A)  $0.28$ 
B)  $3.28$ 
C)  $1.28$ 
D)  $2.28$ 
E)  $1.08$ 
A)  $0.28$ 
B)  $3.28$ 
C)  $1.28$ 
D)  $2.28$ 
E)  $1.08$

Question 8

The velocity graph of a braking car is shown. Use it to estimate to the nearest foot the distance traveled by the car while the brakes are applied.Use a left sum with n = 7.

Accepted Answer

The answer of The velocity graph of a braking car...

Question 9

The velocity graph of a car accelerating from rest to a speed of 7 km/h over a period of 10 seconds is shown. Estimate to the nearest integer the distance traveled during this period. Use a right sum with $n=10$ .

Accepted Answer

The answer of The velocity graph of a car accelerating...

Question 10

Evaluate the integral. $\int_{a / 8}^{s / 6} \sin ~t ~d t$

Accepted Answer

A)  $-0.943$ 
B)  $0.857$ 
C)  $1.057$ 
D)  $0.057$ 
E)  $0.557$ 
A)  $-0.943$ 
B)  $0.857$ 
C)  $1.057$ 
D)  $0.057$ 
E)  $0.557$

Question 11

The velocity of a car was read from its speedometer at ten-second intervals and recorded in the table. Use the Midpoint Rule to estimate the distance traveled by the car. $t(\mathrm{s})$ $v(\mathrm{mi} / \mathrm{h})$ $t(\mathrm{s})$ $v(\mathrm{mi} / \mathrm{h})$ 0 0
60 54 10 $34$ 70 $67$ 20 $43$ 80 $77$ 30 $36$ 90 $37$ 40 $45$ 100 $42$ 50 $45$

Accepted Answer

A)  1.2 miles 
B)  1.8 miles 
C)  0.8 miles 
D)  2.4 miles 
E)  0.6 miles 
A)  1.2 miles 
B)  1.8 miles 
C)  0.8 miles 
D)  2.4 miles 
E)  0.6 miles

Question 12

Evaluate the integral. $\int_{0}^{1} 7 x^{6} \cos \left(x^{7}\right) d x$

Accepted Answer

The answer of Evaluate the integral. \(\int_{0}^{1} 7 x^{6} \cos...

Question 13

Evaluate the integral if it exists. $\int_{0}^{1} 6 x^{2} \cos \left(x^{3}\right) d x$

Accepted Answer

A)  0 
B)  $\cos (3)$ 
C)  $\sin (6)$ 
D)  $3 \sin (3)$ 
E)  none of these 
A)  0 
B)  $\cos (3)$ 
C)  $\sin (6)$ 
D)  $3 \sin (3)$ 
E)  none of these

Question 14

Evaluate the integral by making the given substitution. $\int x^{2} \sqrt{x^{3}+2} d x, \quad \quad u=x^{3}+2$

Accepted Answer

A)  $-\frac{2}{9}\left(x^{3}+2\right)^{3 / 2}+C$ 
B)  $\frac{2}{9}\left(x^{3}+2\right)^{3 / 2}$ 
C)  $\frac{1}{9}\left(x^{3}+2\right)^{1 / 2}+C$ 
D)  $\frac{2}{9}\left(x^{3}+2\right)^{1 / 2}+C$ 
E)  $\frac{2}{9}\left(x^{3}+2\right)^{3 / 2}+C$ 
A)  $-\frac{2}{9}\left(x^{3}+2\right)^{3 / 2}+C$ 
B)  $\frac{2}{9}\left(x^{3}+2\right)^{3 / 2}$ 
C)  $\frac{1}{9}\left(x^{3}+2\right)^{1 / 2}+C$ 
D)  $\frac{2}{9}\left(x^{3}+2\right)^{1 / 2}+C$ 
E)  $\frac{2}{9}\left(x^{3}+2\right)^{3 / 2}+C$

Question 15

Use the definition of area to find the area of the region under the graph of f on [a, b] using the indicated choice of ck. $f(x)$ = $x^{2}$ , [0, 2], ck is the left endpoint

Accepted Answer

Question 16

Approximate the area under the curve $y=\frac{2}{x^{2}}$ from 1 to 2 using ten approximating rectangles of equal widths and right endpoints. Round the answer to the nearest hundredth.

Accepted Answer

The answer of Approximate the area under the curve $y=\frac{2}{x^{2}}$...

Question 17

Evaluate the indefinite integral. $\int 4 x\left(x^{2}+3\right)^{4} d x$

Accepted Answer

The answer of Evaluate the indefinite integral. \(\int 4 x\left(x^{2}+3\right)^{4}...

Question 18

Use the definition of area to find the area of the region under the graph of f on [a, b] using the indicated choice of ck. $f(x)$ = $x^{2}$ + 6x + 1, [ $-1$ , 1], ck is the right endpoint

Accepted Answer

Question 19

Evaluate $\int_{0}^{3}\left(x+\sqrt{25-x^{2}}\right) d x$ by interpreting it in terms of areas.

Accepted Answer

The answer of Evaluate $\int_{0}^{3}\left(x+\sqrt{25-x^{2}}\right) d x$ by interpreting it...

Question 20

Express the integral as a limit of sums. Then evaluate the limit. $\int_{0}^{x} \sin ~13 x ~~d x$

Accepted Answer

The answer of Express the integral as a limit of...

Evaluate by interpreting it in terms of areas. $\int_{-4}^{4} \sqrt{16-x^{2}} ~~d x$

Estimate the area from 0 to 5 under the graph of $f(x)=25-x^{2}$ using five approximating rectangles and right endpoints.

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h(x)=\int_{1}^{e^{x}} 2 \ln t d t$

If $f(x)=\sqrt{x}-3,1 \leq x \leq 6$ , find the Riemann sum with n = 5 correct to 3 decimal places, taking the sample points to be midpoints.

Evaluate the integral. $\int_{0}^{1} \frac{e^{2 x}}{1+e^{4 x}} d x$

Evaluate the definite integral. $\int_{0}^{\pi / 8} \sin~ 5 t ~d t$

The velocity graph of a braking car is shown. Use it to estimate to the nearest foot the distance traveled by the car while the brakes are applied.Use a left sum with n = 7.

The velocity graph of a car accelerating from rest to a speed of 7 km/h over a period of 10 seconds is shown. Estimate to the nearest integer the distance traveled during this period. Use a right sum with $n=10$ .

Evaluate the integral. $\int_{a / 8}^{s / 6} \sin ~t ~d t$

Evaluate the integral. $\int_{0}^{1} 7 x^{6} \cos \left(x^{7}\right) d x$

Evaluate the integral if it exists. $\int_{0}^{1} 6 x^{2} \cos \left(x^{3}\right) d x$

Evaluate the integral by making the given substitution. $\int x^{2} \sqrt{x^{3}+2} d x, \quad \quad u=x^{3}+2$

Use the definition of area to find the area of the region under the graph of f on [a, b] using the indicated choice of c_k. $f(x)$ = $x^{2}$ , [0, 2], c_k is the left endpoint

Approximate the area under the curve $y=\frac{2}{x^{2}}$ from 1 to 2 using ten approximating rectangles of equal widths and right endpoints. Round the answer to the nearest hundredth.

Evaluate the indefinite integral. $\int 4 x\left(x^{2}+3\right)^{4} d x$

Use the definition of area to find the area of the region under the graph of f on [a, b] using the indicated choice of c_k. $f(x)$ = $x^{2}$ + 6x + 1, [ $-1$ , 1], c_k is the right endpoint

Evaluate $\int_{0}^{3}\left(x+\sqrt{25-x^{2}}\right) d x$ by interpreting it in terms of areas.

Express the integral as a limit of sums. Then evaluate the limit. $\int_{0}^{x} \sin ~13 x ~~d x$

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

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Exam 5: Applications of Integration

Evaluate by interpreting it in terms of areas. ∫−4416−x2 dx\int_{-4}^{4} \sqrt{16-x^{2}} ~~d x∫−44​16−x2​ dx

Estimate the area from 0 to 5 under the graph of f(x)=25−x2f(x)=25-x^{2}f(x)=25−x2 using five approximating rectangles and right endpoints.

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. h(x)=∫1ex2ln⁡tdth(x)=\int_{1}^{e^{x}} 2 \ln t d th(x)=∫1ex​2lntdt

If f(x)=x−3,1≤x≤6f(x)=\sqrt{x}-3,1 \leq x \leq 6f(x)=x​−3,1≤x≤6 , find the Riemann sum with n = 5 correct to 3 decimal places, taking the sample points to be midpoints.

Evaluate the integral. ∫01e2x1+e4xdx\int_{0}^{1} \frac{e^{2 x}}{1+e^{4 x}} d x∫01​1+e4xe2x​dx

The area of the region that lies to the right of the y-axis and to the left of the parabola x=6y−y2x=6 y-y^{2}x=6y−y2 (the shaded region in the figure) is given by the integral ∫05(6y−y2)dy\int_{0}^{5}\left(6 y-y^{2}\right) d y∫05​(6y−y2)dy . Find the area..

Evaluate the definite integral. ∫0π/8sin⁡ 5t dt\int_{0}^{\pi / 8} \sin~ 5 t ~d t∫0π/8​sin 5t dt

The velocity graph of a braking car is shown. Use it to estimate to the nearest foot the distance traveled by the car while the brakes are applied.Use a left sum with n = 7.

The velocity graph of a car accelerating from rest to a speed of 7 km/h over a period of 10 seconds is shown. Estimate to the nearest integer the distance traveled during this period. Use a right sum with n=10n=10n=10 .

Evaluate the integral. ∫a/8s/6sin⁡ t dt\int_{a / 8}^{s / 6} \sin ~t ~d t∫a/8s/6​sin t dt

Evaluate the integral. ∫017x6cos⁡(x7)dx\int_{0}^{1} 7 x^{6} \cos \left(x^{7}\right) d x∫01​7x6cos(x7)dx

Evaluate the integral if it exists. ∫016x2cos⁡(x3)dx\int_{0}^{1} 6 x^{2} \cos \left(x^{3}\right) d x∫01​6x2cos(x3)dx

Evaluate the integral by making the given substitution. ∫x2x3+2dx,u=x3+2\int x^{2} \sqrt{x^{3}+2} d x, \quad \quad u=x^{3}+2∫x2x3+2​dx,u=x3+2

Use the definition of area to find the area of the region under the graph of f on [a, b] using the indicated choice of ck. f(x)f(x)f(x) = x2x^{2}x2 , [0, 2], ck is the left endpoint

Approximate the area under the curve y=2x2y=\frac{2}{x^{2}}y=x22​ from 1 to 2 using ten approximating rectangles of equal widths and right endpoints. Round the answer to the nearest hundredth.

Evaluate the indefinite integral. ∫4x(x2+3)4dx\int 4 x\left(x^{2}+3\right)^{4} d x∫4x(x2+3)4dx

Use the definition of area to find the area of the region under the graph of f on [a, b] using the indicated choice of ck. f(x)f(x)f(x) = x2x^{2}x2 + 6x + 1, [ −1-1−1 , 1], ck is the right endpoint

Evaluate ∫03(x+25−x2)dx\int_{0}^{3}\left(x+\sqrt{25-x^{2}}\right) d x∫03​(x+25−x2​)dx by interpreting it in terms of areas.

Express the integral as a limit of sums. Then evaluate the limit. ∫0xsin⁡ 13x dx\int_{0}^{x} \sin ~13 x ~~d x∫0x​sin 13x dx

Functions and Limits

Derivatives

Applications of Differentiation

Integrals

Inverse Functions

Techniques of Integration

Further Applications of Integration

Differential Equations

Parametric Equations and Polar Coordinates

Infinite Sequences and Series

Vectors and the Geometry of Space

Vector Functions

Partial Derivatives

Multiple Integrals

Vector Calculus

Second-Order Differential Equations

Final Exam

Filters

Evaluate by interpreting it in terms of areas. $\int_{-4}^{4} \sqrt{16-x^{2}} ~~d x$

Estimate the area from 0 to 5 under the graph of $f(x)=25-x^{2}$ using five approximating rectangles and right endpoints.

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. $h(x)=\int_{1}^{e^{x}} 2 \ln t d t$

If $f(x)=\sqrt{x}-3,1 \leq x \leq 6$ , find the Riemann sum with n = 5 correct to 3 decimal places, taking the sample points to be midpoints.

Evaluate the integral. $\int_{0}^{1} \frac{e^{2 x}}{1+e^{4 x}} d x$

Evaluate the definite integral. $\int_{0}^{\pi / 8} \sin~ 5 t ~d t$

The velocity graph of a car accelerating from rest to a speed of 7 km/h over a period of 10 seconds is shown. Estimate to the nearest integer the distance traveled during this period. Use a right sum with $n=10$ .

Evaluate the integral. $\int_{a / 8}^{s / 6} \sin ~t ~d t$

Evaluate the integral. $\int_{0}^{1} 7 x^{6} \cos \left(x^{7}\right) d x$

Evaluate the integral if it exists. $\int_{0}^{1} 6 x^{2} \cos \left(x^{3}\right) d x$

Evaluate the integral by making the given substitution. $\int x^{2} \sqrt{x^{3}+2} d x, \quad \quad u=x^{3}+2$

Use the definition of area to find the area of the region under the graph of f on [a, b] using the indicated choice of c_k. $f(x)$ = $x^{2}$ , [0, 2], c_k is the left endpoint

Approximate the area under the curve $y=\frac{2}{x^{2}}$ from 1 to 2 using ten approximating rectangles of equal widths and right endpoints. Round the answer to the nearest hundredth.

Evaluate the indefinite integral. $\int 4 x\left(x^{2}+3\right)^{4} d x$

Use the definition of area to find the area of the region under the graph of f on [a, b] using the indicated choice of c_k. $f(x)$ = $x^{2}$ + 6x + 1, [ $-1$ , 1], c_k is the right endpoint

Evaluate $\int_{0}^{3}\left(x+\sqrt{25-x^{2}}\right) d x$ by interpreting it in terms of areas.

Express the integral as a limit of sums. Then evaluate the limit. $\int_{0}^{x} \sin ~13 x ~~d x$